Move along Line Path Not constantly
See the NLA editor capabilities for this.  Repeat Loop. Let N1 and N2 be positive integers. So action can be repeated N1 times.  Then a period of no movement.  Then again an action can be repeated N2 times.
Consider this as a quick note on NLA possibilities How to generate a full sprite sheet automatically
Tutorials on NLA can be searched here at BSE or yoootooob or similar site on your selected search engine. This is not a tutorial on NLA.
Insert keyframes for position on your Follow Path Constraint position. If the object moving happens to be an organic character then use keyframes on the motion of legs of character.